Finite Group Databasebeta

← #28348301

Subgroups — #28348301

Order 13696.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293318C211yesmaximal, characteristic, complemented
2#20293272C211yesmaximal, characteristic
3#20292982C211yesmaximal, characteristic, complemented
4#20292959C211yesmaximal, characteristic, complemented
5#20292953C211yesmaximal, characteristic, complemented
6#20292927C211yesmaximal, characteristic, complemented
7#20292867C211yesmaximal, characteristic, complemented
8#20292859C211yesmaximal, characteristic, complemented
9#20292849C211yesmaximal, characteristic, complemented
10#20292786C211yesmaximal, characteristic, complemented
11#20292776C211yesmaximal, characteristic, complemented
12C8×D428C211yesmaximal, characteristic, complemented
13#20292620C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15C4×SD16×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic
18C2×#13133808C2211yescharacteristic, complemented
19C2×#13133805C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic, complemented
22#16812103—21no—
23#16812101C2211yescharacteristic, complemented
24#16812099—21no—
25#16812097C2211yescharacteristic
26#16812094—21no—
27#16812093—21no—
28#16812089C2211yescharacteristic
29#16812085C2211yescharacteristic
30#16812085C2211yescharacteristic, complemented
31#16812084C2211yescharacteristic
32#16812083—21no—
33#16812079—21no—
34C4×D428C2211yescharacteristic
35#16812067C2211yescharacteristic, complemented
36#16812066—21no—
37C4⋊C4×D107C2211yescharacteristic
38C2×(C2×D107)⋊C4—21no—
39#16812042C2211yescharacteristic, complemented
40#16812040C2211yescharacteristic, complemented
41C4⋊(C107⋊C8)C2211yescharacteristic, complemented
42#16812034C2211yescharacteristic, complemented
43#16812034C2211yescharacteristic, complemented
44#16812032C2211yescharacteristic, complemented
45#16812031C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812029C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52C2×SD16×C107C2211yescharacteristic
53C4×Q8×C107C2211yescharacteristic, complemented
54C4×D4×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic
58C4×C8×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133817—22no—
61#13133817—41no—
62C2×Dic214C2311yescharacteristic
63C2×D428—21no—
64C2×D428D411yescharacteristic
65C2×D428C2311yescharacteristic
66C2×C22⋊D107—21no—
67C2×C22⋊D107—21no—
68D4×D107—22no—
69D4×D107—41no—
70C22×Dic107—21no—
71C2×C4×D107—21no—
72C2×C4×D107C2311yescharacteristic
73C2×C4×D107C2311yescharacteristic
74#13133808—41no—
75#13133805—41no—
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C8×D107—41no—
79C8×D107—41no—
80C4⋊Dic107—21no—
81C4⋊Dic107D411yescharacteristic
82C4⋊Dic107C2311yescharacteristic
83C4⋊Dic107C2311yescharacteristic
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4D411yescharacteristic
97(C2×D107)⋊C4D411yescharacteristic
98C4×Dic107—21no—
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107C2311yescharacteristic
101C22×C4×C107—21no—
102SD16×C107—22no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C4⋊C4×C107—21no—
106C4⋊C4×C107C2311yescharacteristic
107C4⋊C4×C107C2311yescharacteristic
108C22⋊C4×C107—21no—
109C42×C107—21no—
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112C22×D107—21no—
113C22×D107—42no—
114C22×D107—42no—
115C22×D107—81no—
116C22×D107C2×D411yescharacteristic
117C22×D107C2×D411yescharacteristic
118Q8×C107—21no—
119Q8×C107—21no—
120D4×C107—21no—
121D4×C107#4112yes—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—81no—
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2411yescharacteristic
129C22⋊D107—42no—
130C22⋊D107—42no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—42no—
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137D428—22no—
138D428—21no—
139D428—21no—
140D428—81no—
141C4×D107—21no—
142C4×D107—21no—
143C4×D107—21no—
144C4×D107—21no—
145C4×D107—42no—
146Dic214—21no—
147Dic214—21no—
148C8×C107—21no—
149C8×C107—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—22no—
153C22×C107—41no—
154C22×C107—41no—
155C22×C107C22×D411yescharacteristic
156C2×D107—21no—
157C2×D107—21no—
158C2×D107—21no—
159C2×D107—21no—
160C2×D107—21no—
161C2×D107—41no—
162C2×D107—41no—
163C2×D107—41no—
164C2×D107—84no—
165C2×D107#13512yes—
166C4×C107—21no—
167C4×C107—21no—
168C4×C107—41no—
169C4×C107—41no—
170C4×C107—41no—
171C4×C107C2×#4111yescharacteristic
172C4×C107H(5,𝔽2)11yescharacteristic
173Dic107—21no—
174Dic107—21no—
175Dic107—41no—
176Dic107—41no—
177Dic107—41no—
178C2×C107—42no—
179C2×C107#54611yescharacteristic
180C2×C107C2×#13511yescharacteristic
181C2×C107#57711yescharacteristic
182D107—22no—
183D107—41no—
184D107—81no—
185#3296—1071nomaximal, Sylow
186C107#329611yescharacteristic, complemented, Sylow
187#552—1071no—
188#546—1071no—
189#488—1071no—
190#483—1071no—
191#482—1071no—
192#478—1071no—
193#467—1071no—
194#465—1071no—
195#464—1071no—
196#452—1071no—
197#450—1071no—
198C4×SD16D10711yescharacteristic, complemented
199C8×D4—1071no—
200#416—1071no—
201C2×#102—1071no—
202C2×#41—1071no—
203C22×D4—1071no—
204C2×Q16—1071no—
205C2×SD16—1071no—
206C2×SD16C2×D10711yescharacteristic
207C22×C8—1071no—
208C22×C8—1071no—
209#126—2141no—
210#125—1071no—
211#125—2141no—
212#124—1071no—
213#123—2141no—
214#123—2141no—
215C22⋊Q8—1071no—
216C22⋊Q8—1071no—
217C22⋊Q8—1071no—
218#121—1071no—
219#121—2141no—
220#120—2141no—
221C4×Q8C2×D10711yescharacteristic, complemented
222C4×D4—1071no—
223C4×D4C2×D10711yescharacteristic
224#117—1071no—
225#117—2141no—
226C2×C4⋊C4—1071no—
227C2×C22⋊C4—2141no—
228#107—1071no—
229#106—1071no—
230#106C2×D10711yescharacteristic, complemented
231C4⋊C8—1071no—
232#103—1071no—
233#103—1071no—
234#103—1071no—
235#103C2×D10711yescharacteristic, complemented
236#102—1071no—
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#102—1071no—
241#102C2×D10711yescharacteristic
242C22⋊C8—1071no—
243C22⋊C8—1071no—
244C4×C8C2×D10711yescharacteristic, complemented
245C24—2141no—
246#41—4281no—
247#41—2142no—
248C2×Q8—1071no—
249C2×Q8C22×D10711yescharacteristic
250C2×D4—1071no—
251C2×D4—1071no—
252C2×D4—4281no—
253C2×D4—2141no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4—2142no—
257C2×D4C22×D10711yescharacteristic
258C22×C4—21no—
259C22×C4—1071no—
260C22×C4—1071no—
261C22×C4—2141no—
262C22×C4—2141no—
263Q16—4281no—
264SD16—22no—
265SD16—4281no—
266C2×C8—1071no—
267C2×C8—1071no—
268C2×C8—4281no—
269C2×C8—4281no—
270C2×C8C22×D10711yescharacteristic
271C2×C8C22×D10711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—1071no—
274C4⋊C4—1071no—
275C4⋊C4—1071no—
276C4⋊C4—2141no—
277C4⋊C4—2141no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4—2141no—
282C4⋊C4C22×D10711yescharacteristic
283C4⋊C4C22×D10711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—1071no—
286C22⋊C4—1071no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C42—21no—
295C42—2141no—
296C42C22×D10711yescharacteristic
297C23—21no—
298C23—8561no—
299C23—1071no—
300C23—1071no—
301C23—4282no—
302C23—4282no—
303C23—2141no—
304Q8—21no—
305Q8—21no—
306Q8—2141no—
307Q8—2141no—
308D4—21no—
309D4—8561no—
310D4—4282no—
311D4—4282no—
312D4—2142no—
313D4—2141no—
314D4—2141no—
315D4#1313381812yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—1071no—
321C2×C4—1071no—
322C2×C4—4282no—
323C2×C4—4282no—
324C2×C4—2141no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4D4×D10711yescharacteristic
332C2×C4D4×D10711yescharacteristic
333C2×C4C23×D10711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2141no—
337C8—2141no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—8564no—
342C22—1072no—
343C22—4281no—
344C22—4281no—
345C22—4281no—
346C22—2141no—
347C22—2141no—
348C22—2141no—
349C22—2141no—
350C22—2141no—
351C22C2×D4×D10711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—4281no—
358C4—4281no—
359C4—4281no—
360C4—2141no—
361C4—2141no—
362C4C2×#1313381811yescharacteristic
363C4#1681215511yescharacteristic
364C2—42no—
365C2—8561no—
366C2—4281no—
367C2—2142no—
368C2#2029327311yescharacteristic
369C2#135×D10711yescharacteristic
370C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×SD16×C107C2
socleC22×C107C22×D4
prime radical#20293272C2
hypercenterC4×SD16D107
hypocenterC107#3296
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20293318C2semidirect
3. #20292982C2semidirect
4. #20292959C2semidirect
5. #20292953C2semidirect
6. #20292927C2semidirect
7. #20292867C2semidirect
8. #20292859C2semidirect
9. #20292849C2semidirect
10. #20292786C2semidirect
11. #20292776C2semidirect
12. C8×D428C2semidirect
13. #20292620C2semidirect
14. #102×D107C2semidirect
15. C4×SD16×C107C2semidirect
18. C2×#13133808C22semidirect
21. C2×C8×D107C22semidirect
23. #16812101C22semidirect
30. #16812085C22semidirect
35. #16812067C22semidirect
39. #16812042C22semidirect
40. #16812040C22semidirect
41. C4⋊(C107⋊C8)C22semidirect
42. #16812034C22semidirect
43. #16812034C22semidirect
44. #16812032C22semidirect
51. (C2×D107)⋊C8C22semidirect
53. C4×Q8×C107C22semidirect
55. #106×C107C22semidirect
56. #103×C107C22semidirect
58. C4×C8×C107C22semidirect
186. C107#3296semidirect
198. C4×SD16D107semidirect
221. C4×Q8C2×D107semidirect
230. #106C2×D107semidirect
235. #103C2×D107semidirect
244. C4×C8C2×D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202933182
#202932722
#202929822
#202929592
#202929532
#202929272
#202928672
#202928592
#202928492
#202927862
#202927762
C8×D4282
#202926202
#102×D1072
C4×SD16×C1072
#3296107